Finite Group Databasebeta

← #356065

Subgroups — #356065

Order 512.

3182 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#407C211yesmaximal, characteristic
2C23×C22⋊C8C211yesmaximal, characteristic
3C2×#1752C211yesmaximal, characteristic, complemented
4C2×#1753C212yesmaximal, complemented
5C2×C8×C22⋊C4C211yesmaximal, characteristic, complemented
6C4×#379C211yesmaximal, characteristic
7#10922C212yesmaximal, complemented
8#10921C212yesmaximal, complemented
9#10715C212yesmaximal, complemented
10#10714C211yesmaximal, characteristic, complemented
11#10714C211yesmaximal, characteristic, complemented
12C25×C4C2211yescharacteristic
13C23×M16C2212yes—
14C24×C8C2211yescharacteristic
15C24×C8C2211yescharacteristic
16C2×#407—24no—
17C2×#407—24no—
18C2×#407C418yescomplemented
19C22×C22⋊C8—28no—
20C22×C22⋊C8C418yescomplemented
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C22×C22⋊C8C2211yescharacteristic
24C22×C22⋊C8C2212yes—
25C22×C22⋊C8C2211yescharacteristic
26C22×C22⋊C8C2211yescharacteristic
27C22×C22⋊C8C2211yescharacteristic
28C22×#97C2212yescomplemented
29C22×C4×C8C2212yescomplemented
30C2×#379C412yes—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2212yes—
33C2×C4×C22⋊C4C2211yescharacteristic
34C2×C4×C22⋊C4C2212yes—
35#1753—28no—
36#1752—24no—
37C8×C22⋊C4—24no—
38C2×#336C2212yescomplemented
39C2×#336C2212yescomplemented
40C2×#336C2212yescomplemented
41C2×#336C2212yescomplemented
42C2×#336C2212yescomplemented
43C2×#336C2212yescomplemented
44C22⋊#95C412yes—
45C22⋊#95C412yes—
46C22⋊#95C412yes—
47C26C2×C411yescharacteristic
48C24×C4—24no—
49C24×C4—22no—
50C24×C4—22no—
51C24×C4—22no—
52C24×C4—22no—
53C24×C4—24no—
54C24×C4—42no—
55C24×C4—42no—
56C24×C4C2×C411yescharacteristic
57C24×C4C2×C411yescharacteristic
58C24×C4C2×C411yescharacteristic
59C24×C4C2×C414yes—
60C24×C4C2×C414yes—
61C24×C4C2311yescharacteristic
62C24×C4C2311yescharacteristic
63C24×C4C2311yescharacteristic
64C22×M16—24no—
65C22×M16—24no—
66C22×M16—24no—
67C22×M16—24no—
68C22×M16—42no—
69C22×M16—42no—
70C22×M16C2×C418yes—
71C23×C8—22no—
72C23×C8—22no—
73C23×C8—22no—
74C23×C8—22no—
75C23×C8—24no—
76C23×C8—24no—
77C23×C8—42no—
78C23×C8—42no—
79C23×C8C2×C414yes—
80C23×C8C2×C414yes—
81C23×C8C2311yescharacteristic
82C23×C8C2311yescharacteristic
83C23×C8C2312yes—
84C23×C8C2311yescharacteristic
85C23×C8C2312yes—
86C23×C8C2311yescharacteristic
87C22×C22⋊C4C2×C412yes—
88C22×C22⋊C4C2×C414yes—
89C22×C22⋊C4C2×C414yes—
90C22×C22⋊C4C2×C412yes—
91C22×C42C2312yes—
92C22×C42C2312yes—
93#407—232no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8—24no—
98C2×C22⋊C8—24no—
99C2×C22⋊C8—24no—
100C2×C22⋊C8—24no—
101C2×C22⋊C8—232no—
102C2×C22⋊C8C2×C414yes—
103C2×C22⋊C8C2×C414yes—
104C2×C22⋊C8C2×C414yes—
105C2×C22⋊C8C2×C414yes—
106C2×C22⋊C8C2×C418yes—
107C2×C22⋊C8C2×C414yes—
108C2×C22⋊C8C2×C414yes—
109C2×C22⋊C8D418yes—
110C2×C22⋊C8D414yes—
111C2×C22⋊C8D414yes—
112C2×#97—24no—
113C2×#97—24no—
114C2×#97—42no—
115C2×C4×C8—24no—
116C2×C4×C8—24no—
117C2×C4×C8—42no—
118#379—28no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4—24no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—24no—
125C4×C22⋊C4—42no—
126C4×C22⋊C4—42no—
127C2×#95C2×C412yes—
128C2×#95C2×C414yes—
129C2×#95C2×C412yes—
130C2×#95C2×C414yes—
131#336—24no—
132#336—24no—
133#336—24no—
134#336—24no—
135#336—42no—
136#336—42no—
137C25—24no—
138C25—24no—
139C25—24no—
140C25—22no—
141C25—22no—
142C25—24no—
143C25—42no—
144C25—42no—
145C25C4214yes—
146C25C22×C411yescharacteristic
147C25C22×C411yescharacteristic
148C25C22×C411yescharacteristic
149C23×C4—22no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—22no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—22no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—24no—
160C23×C4—22no—
161C23×C4—22no—
162C23×C4—28no—
163C23×C4—28no—
164C23×C4—28no—
165C23×C4—28no—
166C23×C4—24no—
167C23×C4—22no—
168C23×C4—22no—
169C23×C4—28no—
170C23×C4—28no—
171C23×C4—216no—
172C23×C4—24no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—22no—
176C23×C4—21no—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—22no—
181C23×C4—22no—
182C23×C4—22no—
183C23×C4—44no—
184C23×C4—44no—
185C23×C4—48no—
186C23×C4—42no—
187C23×C4—42no—
188C23×C4—42no—
189C23×C4—48no—
190C23×C4—48no—
191C23×C4—48no—
192C23×C4—48no—
193C23×C4—48no—
194C23×C4—48no—
195C23×C4—48no—
196C23×C4—48no—
197C23×C4C4214yes—
198C23×C4C4214yes—
199C23×C4C4214yes—
200C23×C4C22×C414yes—
201C23×C4C22×C411yescharacteristic
202C23×C4C22×C412yes—
203C23×C4C22×C412yes—
204C23×C4C22×C414yes—
205C23×C4C22×C411yescharacteristic
206C23×C4C22×C411yescharacteristic
207C23×C4C22×C412yes—
208C23×C4C22×C412yes—
209C23×C4C22×C412yes—
210C23×C4C22×C412yes—
211C23×C4C2×D412yes—
212C23×C4C2×D412yes—
213C23×C4C2×D414yes—
214C23×C4#4112yes—
215C23×C4#4112yes—
216C23×C4#4114yes—
217C23×C4C2411yescharacteristic
218C2×M16—216no—
219C2×M16—216no—
220C2×M16—48no—
221C2×M16—48no—
222C2×M16—48no—
223C2×M16—48no—
224C2×M16—48no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—22no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—22no—
231C22×C8—22no—
232C22×C8—22no—
233C22×C8—22no—
234C22×C8—22no—
235C22×C8—28no—
236C22×C8—28no—
237C22×C8—21no—
238C22×C8—21no—
239C22×C8—22no—
240C22×C8—22no—
241C22×C8—22no—
242C22×C8—22no—
243C22×C8—22no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—22no—
247C22×C8—22no—
248C22×C8—22no—
249C22×C8—22no—
250C22×C8—22no—
251C22×C8—28no—
252C22×C8—28no—
253C22×C8—44no—
254C22×C8—44no—
255C22×C8—48no—
256C22×C8—48no—
257C22×C8—48no—
258C22×C8—48no—
259C22×C8C22×C412yes—
260C22×C8C22×C412yes—
261C22×C8C22×C414yes—
262C22×C8C22×C412yes—
263C22×C8C22×C412yes—
264C22×C8C22×C414yes—
265C22×C8C2×D414yes—
266C22×C8C2×D414yes—
267C22×C8C2×D414yes—
268C22×C8C2×D414yes—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4M1618yescomplemented
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C22⋊C8—28no—
294C22⋊C8—28no—
295C22⋊C8—28no—
296C22⋊C8—28no—
297C22⋊C8—216no—
298C22⋊C8C22⋊C4116yescomplemented
299C22⋊C8C22⋊C4116yescomplemented
300#97—48no—
301C4×C8—48no—
302#95—24no—
303#95—28no—
304#95—24no—
305#95—44no—
306C24—22no—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—232no—
311C24—21no—
312C24—21no—
313C24—22no—
314C24—21no—
315C24—21no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—22no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—22no—
324C24—22no—
325C24—28no—
326C24—28no—
327C24—28no—
328C24—28no—
329C24—28no—
330C24—44no—
331C24—44no—
332C24—48no—
333C24—48no—
334C24—48no—
335C24—48no—
336C24—416no—
337C24—416no—
338C24—416no—
339C24—48no—
340C24—48no—
341C24C2×C4212yes—
342C24C2×C4212yes—
343C24C2×C4212yes—
344C24C4×D414yes—
345C24C4×D412yes—
346C24C4×D412yes—
347C24C2×M1614yes—
348C24#13114yes—
349C24#13114yes—
350C24C23×C411yescharacteristic
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—28no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—416no—
424C22×C4—416no—
425C22×C4—432no—
426C22×C4—432no—
427C22×C4—416no—
428C22×C4—416no—
429C22×C4—48no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—42no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4C2×C4212yes—
442C22×C4C2×C4212yes—
443C22×C4C2×C4212yes—
444C22×C4C2×C22⋊C414yes—
445C22×C4C2×C22⋊C414yes—
446C22×C4#11714yes—
447C22×C4#11714yes—
448C22×C4C4×D414yes—
449C22×C4C4×D414yes—
450C22×C4C4×D418yes—
451C22×C4C4×D412yes—
452C22×C4C4×D412yes—
453C22×C4C4×D414yes—
454C22×C4C2×M1618yes—
455C22×C4C23×C411yescharacteristic
456C22×C4C23×C411yescharacteristic
457C22×C4C22×D411yescharacteristic
458C22×C4C22×D411yescharacteristic
459C22×C4C22×D412yes—
460C22×C4C2×#4111yescharacteristic
461C22×C4C2×#4111yescharacteristic
462C22×C4C2×#4112yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)12yes—
465M16—432no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—24no—
472C2×C8—24no—
473C2×C8—24no—
474C2×C8—24no—
475C2×C8—24no—
476C2×C8—24no—
477C2×C8—24no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—416no—
490C2×C8—416no—
491C2×C8—42no—
492C2×C8—42no—
493C2×C8—42no—
494C2×C8—42no—
495C2×C8—42no—
496C2×C8—42no—
497C2×C8—42no—
498C2×C8—42no—
499C2×C8C2×C22⋊C414yes—
500C2×C8C2×C22⋊C414yes—
501C2×C8C2×C22⋊C418yes—
502C22⋊C4—216no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C42—42no—
507C42—42no—
508C42—42no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—42no—
513C42—42no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—22no—
518C23—28no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—22no—
525C23—22no—
526C23—232no—
527C23—24no—
528C23—24no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—28no—
537C23—28no—
538C23—24no—
539C23—24no—
540C23—28no—
541C23—28no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—416no—
548C23—416no—
549C23—416no—
550C23—416no—
551C23—416no—
552C23—464no—
553C23—464no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—416no—
567C23C4×C22⋊C414yes—
568C23C4×C22⋊C414yes—
569C23C4×M1614yes—
570C23#40514yes—
571C23#40514yes—
572C23#43518yes—
573C23#43618yes—
574C23C22×C4211yescharacteristic
575C23C2×C4×D411yescharacteristic
576C23C2×C4×D411yescharacteristic
577C23C2×C4×D412yes—
578C23#51812yes—
579C23C22×M1612yes—
580C23C2×#13112yes—
581C23C2×#13112yes—
582C23#56812yes—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—416no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4C4×C22⋊C414yes—
632C2×C4C4×C22⋊C414yes—
633C2×C4C22×C22⋊C411yescharacteristic
634C2×C4C22×C22⋊C411yescharacteristic
635C2×C4C2×#11711yescharacteristic
636C2×C4C2×#11711yescharacteristic
637C2×C4C2×C4×D411yescharacteristic
638C2×C4C2×C4×D411yescharacteristic
639C2×C4C2×C4×D412yes—
640C2×C4#51811yescharacteristic
641C2×C4#51811yescharacteristic
642C2×C4#51812yes—
643C2×C4D4212yes—
644C2×C4D4212yes—
645C2×C4#54612yes—
646C2×C4#54612yes—
647C2×C4#54612yes—
648C2×C4#54612yes—
649C2×C4#55112yes—
650C2×C4#55112yes—
651C8—24no—
652C8—24no—
653C8—28no—
654C8—42no—
655C8—42no—
656C8—44no—
657C22—24no—
658C22—24no—
659C22—216no—
660C22—216no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—24no—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—464no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—48no—
676C22—48no—
677C22—48no—
678C22—48no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22#175318yescomplemented
685C22C2×C4×C22⋊C411yescharacteristic
686C22C2×C4×C22⋊C411yescharacteristic
687C22#227311yescharacteristic
688C22#235812yes—
689C22#235812yes—
690C22C2×C4×M1611yescharacteristic
691C22C2×#40511yescharacteristic
692C22C2×#40511yescharacteristic
693C22#287311yescharacteristic
694C22C2×#43512yes—
695C22C2×#43612yes—
696C22#292914yes—
697C22#293312yes—
698C22#293312yes—
699C22D4×M1614yes—
700C22#297414yes—
701C22#297614yes—
702C4—24no—
703C4—24no—
704C4—48no—
705C4—42no—
706C4—84no—
707C4D4×C22⋊C411yescharacteristic
708C4D4×C22⋊C411yescharacteristic
709C4D4×C22⋊C412yes—
710C4#234212yes—
711C4#234211yescharacteristic
712C4#234211yescharacteristic
713C2—28no—
714C2—44no—
715C2—44no—
716C2C22⋊C4211yescharacteristic
717C2C2×#175311yescharacteristic
718C2#2038511yescharacteristic
719C2C22⋊C4×M1611yescharacteristic
720C2#2038911yescharacteristic
721C2#2052511yescharacteristic
722C2#2052711yescharacteristic
723C2#2284614yes—
724C2#2284814yes—
subgroupG/H
center, Frattini, cosocleC23×C4C24
commutatorC23C22×C42
socleC24C23×C4
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#1752C2semidirect
4. C2×#1753C2semidirect
5. C2×C8×C22⋊C4C2semidirect
7. #10922C2semidirect
8. #10921C2semidirect
9. #10715C2semidirect
10. #10714C2semidirect
11. #10714C2semidirect
18. C2×#407C4semidirect
20. C22×C22⋊C8C4semidirect
28. C22×#97C22semidirect
29. C22×C4×C8C22semidirect
38. C2×#336C22semidirect
39. C2×#336C22semidirect
40. C2×#336C22semidirect
41. C2×#336C22semidirect
42. C2×#336C22semidirect
43. C2×#336C22semidirect
280. C2×C22⋊C4M16semidirect
298. C22⋊C8C22⋊C4semidirect
299. C22⋊C8C22⋊C4semidirect
684. C22#1753semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#4072
C23×C22⋊C82
C2×#17522
C2×#17532
C2×C8×C22⋊C42
C4×#3792
#109222
#109212
#107152
#107142